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Developed in Consultation with Virginia Educators Table of Contents Virginia Standards of Learning Correlation Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Virginia Standards of Learning Chapter 1 Number and Number Sense Lesson 1 Use the Order of Operations to Simplify Numerical Expressions . . . . . . . . 10 Lesson 2 8.1a Use Properties of Operations to Simplify Numerical Expressions . . . . . . . . 14 8.1a Lesson 3 Scientific Notation . . . . . . . . . . . . . . . . . . . 19 8.1b Lesson 4 Compare and Order Numbers. . . . . . . . . . 24 8.1b Lesson 5 Real Numbers . . . . . . . . . . . . . . . . . . . . . . 28 8.2 Chapter 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Chapter 2 Computation and Estimation Lesson 6 Solve Real-World Problems. . . . . . . . . . . . 38 8.3a, 8.3b Lesson 7 Evaluate Algebraic Expressions . . . . . . . . 45 8.4 Lesson 8 Perfect Squares and Square Roots. . . . . . . . . . . . . . . . . . . . . . . 50 8.5a, 8.5b Chapter 3 Measurement Lesson 9 Vertical, Supplementary, and Complementary Angles . . . . . . . . . . . . . . . 60 8.6a, 8.6b Lesson 10 Solve Volume Problems. . . . . . . . . . . . . . . 67 8.7a Lesson 11 Solve Surface Area Problems . . . . . . . . . . 74 8.7a Lesson 12 Effects of Changing Dimensions . . . . . . . . 81 8.7b Chapter 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Chapter 4 Geometry Lesson 13 Transformations . . . . . . . . . . . . . . . . . . . . . 94 8.8a, 8.8b Lesson 14 Construct a Three-Dimensional Model . . 102 8.9 Lesson 15 The Pythagorean Theorem . . . . . . . . . . . 108 8.10a, 8.10b Lesson 16 Area and Perimeter of Composite Figures . . . . . . . . . . . . . . . . . 113 8.11 Chapter 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4 Duplicating any part of this book is prohibited by law. Chapter 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Virginia Standards of Learning Chapter 5 Probability and Statistics Lesson 17 Probability . . . . . . . . . . . . . . . . . . . . . . . . 126 Lesson 18 8.12 Picture Graphs, Bar Graphs, and Line Graphs . . . . . . . . . . . . . . . . . . . . . . . 133 8.13a Frequency Distributions and Histograms. . . . . . . . . . . . . . . . . . . . . . . . 140 8.13a Lesson 20 Circle Graphs . . . . . . . . . . . . . . . . . . . . . . 147 8.13a Lesson 21 Scatterplots . . . . . . . . . . . . . . . . . . . . . . . 154 8.13b Lesson 19 Chapter 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Chapter 6 Patterns, Functions, and Algebra Lesson 22 Represent Relations. . . . . . . . . . . . . . . . . 166 8.14 Lesson 23 Represent Functions . . . . . . . . . . . . . . . . 172 8.14, 8.17 Lesson 24 Solve Linear Equations . . . . . . . . . . . . . . 178 8.15a, 8.15c Lesson 25 Solve Linear Inequalities . . . . . . . . . . . . . 185 8.15b Lesson 26 Graph Linear Equations . . . . . . . . . . . . . . 191 8.16, 8.17 Chapter 6 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Pretest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Posttest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Duplicating any part of this book is prohibited by law. Grade 8 Mathematics Formula Sheet . . . . . . . . . . . . 253 5 Chapter 1 t -FTTPO Standards of Learning: 8.1a Use the Order of Operations to Simplify Numerical Expressions Getting the Idea You can simplify a numerical expression that contains more than one operation by using the order of operations. The order of operations tells you the order in which to perform the operations in an expression. Order of Operations 1. Perform operations within grouping symbols, such as parentheses, brackets, or braces. 2. Perform calculations involving exponents or square roots. 3. Multiply and divide from left to right. 4. Add and subtract from left to right. Example 1 1. Simplify the expression 8 2 __ 8 Strategy Step 1 Follow the order of operations. There are no grouping symbols or exponents. Perform multiplication and division from left to right. Multiply. 1 8 2 __ Step 2 Perform addition and subtraction from left to right. Subtract. 1 8 __ Solution 10 4 3 __ 7 4 3 1 ⴝ 7__ 8 ⴚ 2 ⴛ __ 4 8 Duplicating any part of this book is prohibited by law. 8 2 8 __ 8 __ 81 4 Example 2 Simplify: [(5 2) 10] 4 2 Strategy Step 1 Follow the order of operations. There are two grouping symbols: parentheses and brackets. First, apply the order of operations inside the parentheses. Clear the exponent. Then divide. 2 [(5 2) 10] 4 [(25 2) 10] 4 [12.5 10] 4 Step 2 Perform the operation within the brackets. Subtract. [12.5 10] 4 2.5 4 Step 3 Multiply and divide from left to right. Multiply. 2.5 4 10 Solution [(52 ⴜ 2) ⴚ 10] ⴛ 4 ⴝ 10 In addition to parentheses, brackets, and braces, the absolute value symbol, fraction bar, and square root sign are also treated as grouping symbols. Duplicating any part of this book is prohibited by law. 11 Example 3 8 ______ Simplify: 22 6 10 5 Strategy Step 1 Follow the order of operations. Remember that the fraction bar is a grouping symbol. Simplify the numerator of the fraction. 22 8 ______ 6 10 5 30 ___ 6 10 5 Step 2 Multiply and divide from left to right. 30 ___ 6 10 5 6 6 10 6 60 Step 3 Add and subtract from left to right. Add. 6 60 66 Solution 22 ⴙ8 ______ ⴙ 6 ⴛ 10 ⴝ 66 5 Coached Example Simplify the expression 72 ⴚ 3(25 ⴚ 6). Follow the order of operations. 2 Evaluate. 7 3(25 6) ___________ What is the second step? Evaluate. ___________ What is the third step? Evaluate. ___________ What is the fourth step? Evaluate. ___________ 2 7 ⴚ 3(25 ⴚ 6) ⴝ ___________ t $IBQUFSNumber and Number Sense Duplicating any part of this book is prohibited by law. What is the first step? Lesson 1: Use the Order of Operations to Simplify Numerical Expressions Lesson Practice Choose the correct answer. 1. 5. Simplify the following expression. 92 (12 8) __________ 30 10 2 12_12 2 21 A. 14 B. 29 2. C. 128 D. 604 Simplify the following expression. 6. 45 23 ______ 3(5 8) 10 A. 17.2 B. 3.7 3. 7. C. 10.5 D. 15 D. 40_1 5 Simplify the following expression. C. 49 D. 62 Simplify the following expression. A. 135 B. 71 8. 6217 Duplicating any part of this book is prohibited by law. 9. 7 B. 37__ 10 42 11 5 Simplify the following expression. C. 39 D. 25 Simplify the following expression. (29 14) 3 2 A. 1_17 C. 6_27 4_37 D. 8_17 B. C. 39 A. 11 B. 37 9 12 _12 4. 1 A. 36__ 10 24 (12 31) 18 (3) C. 5.7 D. 36.8 Simplify the following expression. A. 3 B. 1.5 Simplify the following expression. A. 75 C. B. 16_23 D. 55_23 15 Fran is simplifying the following expression: _1 52 (7 4 2). 5 A. According to the order of operations, what operation should be performed first to simplify the expression? B. Simplify the expression. Show your work. Answer: 13